Question

here are the endpoints of the segments jk, mn, and pq.
j(-2, 1), k(3, 4)
m(-6, 1), n(-7, 7)
p(-5, -8), q(-6, -2)
follow the directions below.
(a) find the length of each segment.
give an exact answer (not a decimal approximation).
jk =
mn =
pq =
(b) check all statements that are true below.

Explanation:

Part (a)

To find the length of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

For segment \(JK\) with \(J(-2, 1)\) and \(K(3, 4)\):

Step 1: Identify coordinates

\(x_1 = -2\), \(y_1 = 1\), \(x_2 = 3\), \(y_2 = 4\)

Step 2: Apply distance formula

\[

$$\begin{align*} JK&=\sqrt{(3 - (-2))^2 + (4 - 1)^2}\\ &=\sqrt{(5)^2 + (3)^2}\\ &=\sqrt{25 + 9}\\ &=\sqrt{34} \end{align*}$$

\]

For segment \(MN\) with \(M(-6, 1)\) and \(N(-7, 7)\):

Step 1: Identify coordinates

\(x_1 = -6\), \(y_1 = 1\), \(x_2 = -7\), \(y_2 = 7\)

Step 2: Apply distance formula

\[

$$\begin{align*} MN&=\sqrt{(-7 - (-6))^2 + (7 - 1)^2}\\ &=\sqrt{(-1)^2 + (6)^2}\\ &=\sqrt{1 + 36}\\ &=\sqrt{37} \end{align*}$$

\]

For segment \(PQ\) with \(P(-5, -8)\) and \(Q(-6, -2)\):

Step 1: Identify coordinates

\(x_1 = -5\), \(y_1 = -8\), \(x_2 = -6\), \(y_2 = -2\)

Step 2: Apply distance formula

\[

$$\begin{align*} PQ&=\sqrt{(-6 - (-5))^2 + (-2 - (-8))^2}\\ &=\sqrt{(-1)^2 + (6)^2}\\ &=\sqrt{1 + 36}\\ &=\sqrt{37} \end{align*}$$

\]

Part (b) (assuming common true statements based on lengths)

If we check statements about congruence (segments with equal length are congruent):

• \(MN\) and \(PQ\) both have length \(\sqrt{37}\), so " \(MN \cong PQ\)" is true.
• \(JK\) has length \(\sqrt{34}\), different from \(MN\) and \(PQ\), so " \(JK \cong MN\)" and " \(JK \cong PQ\)" are false.

Final Answers (Part a)

\(JK = \boldsymbol{\sqrt{34}}\)
\(MN = \boldsymbol{\sqrt{37}}\)
\(PQ = \boldsymbol{\sqrt{37}}\)

(For part b, the true statement would be \(MN \cong PQ\) depending on the given options, but since options are not fully visible, the length calculations are as above.)

Answer:

Part (a)

To find the length of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

For segment \(JK\) with \(J(-2, 1)\) and \(K(3, 4)\):

Step 1: Identify coordinates

\(x_1 = -2\), \(y_1 = 1\), \(x_2 = 3\), \(y_2 = 4\)

Step 2: Apply distance formula

\[

$$\begin{align*} JK&=\sqrt{(3 - (-2))^2 + (4 - 1)^2}\\ &=\sqrt{(5)^2 + (3)^2}\\ &=\sqrt{25 + 9}\\ &=\sqrt{34} \end{align*}$$

\]

For segment \(MN\) with \(M(-6, 1)\) and \(N(-7, 7)\):

Step 1: Identify coordinates

\(x_1 = -6\), \(y_1 = 1\), \(x_2 = -7\), \(y_2 = 7\)

Step 2: Apply distance formula

\[

$$\begin{align*} MN&=\sqrt{(-7 - (-6))^2 + (7 - 1)^2}\\ &=\sqrt{(-1)^2 + (6)^2}\\ &=\sqrt{1 + 36}\\ &=\sqrt{37} \end{align*}$$

\]

For segment \(PQ\) with \(P(-5, -8)\) and \(Q(-6, -2)\):

Step 1: Identify coordinates

\(x_1 = -5\), \(y_1 = -8\), \(x_2 = -6\), \(y_2 = -2\)

Step 2: Apply distance formula

\[

$$\begin{align*} PQ&=\sqrt{(-6 - (-5))^2 + (-2 - (-8))^2}\\ &=\sqrt{(-1)^2 + (6)^2}\\ &=\sqrt{1 + 36}\\ &=\sqrt{37} \end{align*}$$

\]

Part (b) (assuming common true statements based on lengths)

If we check statements about congruence (segments with equal length are congruent):

• \(MN\) and \(PQ\) both have length \(\sqrt{37}\), so " \(MN \cong PQ\)" is true.
• \(JK\) has length \(\sqrt{34}\), different from \(MN\) and \(PQ\), so " \(JK \cong MN\)" and " \(JK \cong PQ\)" are false.

Final Answers (Part a)

\(JK = \boldsymbol{\sqrt{34}}\)
\(MN = \boldsymbol{\sqrt{37}}\)
\(PQ = \boldsymbol{\sqrt{37}}\)

(For part b, the true statement would be \(MN \cong PQ\) depending on the given options, but since options are not fully visible, the length calculations are as above.)